Binary conversion practice — base-10 to base-2 Convert the decimal number 213 (base 10) into its binary representation (base 2).

Introduction / Context:
Converting from decimal to binary is fundamental in digital systems. The idea is to express the same quantity using powers of 2. There are two common manual methods: repeated subtraction of powers of 2 and repeated division by 2. We will apply the powers-of-2 method to convert 213 to binary cleanly and verify it afterward.

Given Data / Assumptions:

• Input: 213 in base 10.
• Unsigned representation; width is minimal (no fixed 8/16-bit constraint unless formatting options require).
• We will present an 8-bit form to match the options.

Concept / Approach:

Find the largest power of 2 ≤ number, set that bit, subtract, and continue. Powers of 2 near 213 are 128 (2^7), 64 (2^6), 32 (2^5), 16 (2^4), 8 (2^3), 4 (2^2), 2 (2^1), 1 (2^0). Set a bit to 1 when using that power, else 0.

Step-by-Step Solution:

Verification / Alternative check:

Evaluate 11010101 as decimal: 128 + 64 + 16 + 4 + 1 = 213. Match confirmed.

Why Other Options Are Wrong:

11001101 = 128 + 64 + 8 + 4 + 1 = 205.

01111001 = 64 + 32 + 16 + 8 + 1 = 121.

11100011 = 128 + 64 + 32 + 2 + 1 = 227.

Common Pitfalls:

Forgetting a power, reversing bit order, or assuming fixed-width without matching the option formatting. Always re-sum to double-check the binary pattern.

Final Answer:

11010101